## 发光学(荧光粉)进阶版

• S-A距离小于0.8 nm，那么就是exchange interaction

## 荧光粉晶体场理论和Dorenbos model

(1) 对于Eu2+也可以画类似的图形。
The effect of the host crystal on this energy difference is expressed by the redshift $$D$$ and the Stokes shift $$\Delta S$$. The energy of f→d absorption and of the d→f emission can be written respectively, as $$E_{\mathrm{abs}}=E_{\mathrm{free}}-D$$ and $$E_{\text {em }}=E_{\text {free }}-D-\Delta S$$. $$E_{\text {free }}$$(Eu2+) = 4.19 eV  is the energy of the f→d transition for free (gaseous) Eu2+ ions. 参考Luminescent characterization of CaAl2S4:Eu powder-JL-2007

(2) Eu2+和Ce3+的CFS差异：[Suppression of Eu2+ Luminescence Loss-AOM-2022] Literally, the CFS energy of 5d state is quantified as the energy difference of 5d1 and 5d5 compo-nents, while because of the heavy overlap between different excitations of Eu2+, this process of evaluation is restricted. In com-parison, Ce3+ owns the explicit distribution of different 4f-5d1-5 transitions in the excitation spectrum

OL6其中O表示中心离子，L表示配体，optical transitions are localized on the central ion O and the surrounding ligands. 对于一个掺杂离子来说，吸收和发射都会受到基质晶格很大的影响，这是因为掺杂离子O的energy level is altered by the ineraction with the neighboring ions L，这种相互作用其实就是nature of the chemcial bonds between O and L:
(1) covalency (共价性)
(2) bond length (键长)
(3) coordination number(配位数)
(4) symmetry (对称性)

### Centroid shift

(determined by the so-called nephelauxetic effect)  $$\varepsilon_{c}$$

(1) 随着bond的covalency(共价性)的增大而增大，可以解释为键的共价性越强，那么化学键越稳定，5d orbital能量越低。
(2) 配位阴离子的polarizability越大，electronegativity越小，那么bond的covalency越大(解释见块引用)
(3) nephelauxetic effect (电子云扩展效应)的含义：The name ‘nephelauxetic’ originates from the Greek for ‘cloud expanding,’ which refers to increased delocalization (expansion) of the dopant d orbital (‘electron cloud’) as a result of increased covalency.
(4) nephelauxetic series $$F^{-}<O^{2-}<C l^{-}<N^{3-} \approx B r^{-}<I^{-}<S^{2-}<S e^{2-}$$

### Crystal field splitting of the 5d orbital

(1) 影响CFS的magnitude的因素有：键长，键的共价性，activator site的配位环境、对称性以及distortion
(2) 键长越长，或者配位多面体体积越大，那么CFS就会减小，对于a simple point charge model，the measure of the crystal field strength$$D_{\mathrm{q}}=\displaystyle \frac{z e^{2} r^{4}}{6 R^{5}}$$，其中$$R$$是离子距离，$$z$$是配位离子的charge或价态，$$r$$是d 波函数的半径。对于octahedral coordination，CFS是$$10D_{\mathrm{q}}$$，也就是劈裂后最高能级到最低能级的能量差。
(3) In cubic symmetry the 5d state splits into two levels (eg and t2g) that will split further for deviations from cubic symmetry. 其中eg劈裂成两个，t2g劈裂成三个。上图中的5d1和5d2的bary center应该就是对应于eg，5d3到5d5的bary center就对应于t2g。对于YAG:Ce3+来说，eg劈裂值$$\Delta_{12}$$，也就是5d1和5d2的差值很大(大约一个电子伏特)，所以Ce3+的发光波长比较长。
(4) CFS变化趋势，octahedral coordination最大，cubic coordination次之， dodecahedral coordination最小。
(5) 激活离子格位的distortion也会影响CFS，比如在YAG:Ce3+中，影响eg劈裂值$$\Delta_{12}$$，从而影响最后的发光峰。
In fact, in the garnet structure the distortion from cubic symmetry is important in explaining the unusually long wavelength for the Ce3+ ion.
(6) bary center

cubic symmetry是什么?YAG

Red shift(Ce3+/Eu2+)自由气态离子最低激发态到基质晶格环境下的最低激发态之间差值。来源有三个
(1) spin-orbit splitting
(2) centroid shift
(3) crystal field splitting

(1) Ce3+-Doped garnet phosphors: composition modification, luminescence properties and applications
(2) 待整理
(1) Introduction to Crystal Field Theory
(2) 晶体场理论简明读本-Chemistry: the Central Science. 13th Edition, Chapter 23.6

Ziazag的形状：It reflects the strength of electron binding within the 4f shell as described by the Jorgensen's spin pairing theory.
Ziazag的形状-不随基质变化：Curves 1 and 2 show a characteristic pattern which is directly related to the strength of the binding of the 4f electrons. Since the 4f electrons are shielded by the filled 5s2 and 5p6 orbitals this pattern is almost independent of the host material. However, the host material does influence the locations of the (whole) curves 1 and 2 relative to the conduction band and the valence band and relative to each other. [Lanthanide energy levels in YPO4-RM-2008]

## 荧光粉光谱项/耦合/选择定则

### 等价电子/不等价电子的的光谱项

Number of states的计算公式为：$$\displaystyle\frac {[2\times (2l+1)]!}{[2\times (2l+1)- q]!q!}$$

### 角动量耦合/自旋耦合

$$L_1$$和$$L_2$$分别表示是以$$l_1$$和$$l_2$$为量子数的角动量，它们的数值分别是：$$\left.\begin{array}{l} L_{1}=\sqrt{l_{1}\left(l_{1}+1\right)} \hbar \\ L_{2}=\sqrt{l_{2}\left(l_{2}+1\right)} \hbar \end{array}\right\}$$把两个角动量加起来：$$L_{1}+L_{2}=L$$显然$$L$$也是角动量，它的数值也满足$$L=\sqrt{l(l+1)} \hbar$$而$$l$$只能有如下取值：$$l=l_{1}+l_{2}, l_{1}+l_{2}-1, \cdots,\left|l_{1}-l_{2}\right|$$为什么$$l$$的取值是这样的？举个例子说明。角动量相加，其实是投影值相加，图中$$m_{l_{1}}$$表示轨道量子数$$l_1=1$$在$$z$$方向的投影取值。

### $$LS$$耦合和$$jj$$耦合

(1) $$G_1$$和$$G_2$$占优($$LS$$耦合)：两个自旋运动合成一个总自旋运动，两个轨道角动量合成一个轨道总角动量，然后轨道总角动量和自旋总角动量合成总角动量。

(2) $$G_3$$和$$G_4$$占优($$jj$$耦合)：电子的自旋同自己的轨道运动比其余几种要强，这时电子的自旋角动量和轨道角动量要先合成各自的总角动量，然后这两个电子的总角动量又合成原子的总角动量。

### 弱/中等/强耦合

(1) $$N$$是未满壳层的电子总个数，$$\Delta=\nabla^{2}=\displaystyle\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}$$
(2) 核电荷因内层电子的屏蔽作用而取$$Z^{*}$$。
(3) 电子间的相互作用$$H_c$$。
(4) 自选轨道耦合$$H_{SO}$$，$$\zeta\left(r_{i}\right)$$是耦合系数(决定耦合的强度)。

(1) $$H_{SO}<<H_{c}$$，弱耦合，一般是原子序数小于30的元素，也叫作 Russell-Saunders 耦合，对应于$$LS$$耦合。比如Mn2+的最低激发态是$${ }^{4} G_{11 / 2}$$， 实际上，这里$$J$$的值还可以取9/2， 7/2， 5/2。但由于$$H_{SO}$$小，即旋轨耦合弱，不同$$J$$值对 应的能量相差不大，通常只简单写成$${ }^{4} G$$，即这个态对$$J$$是简并的
(3) $$H_{SO}>>H_{c}$$，强耦合，对应$$j-j$$耦合。
(4) $$H_{SO}\approx H_{c}$$，中等耦合。比如稀土离子就是此类，$$J$$值对能量的影响不能忽略。比如Pr3+的$${ }^{3} H_{4}$$和$${ }^{3} H_{5}$$相差2000 cm-1

### 重原子效应

Heavy Atom Effect (IUPAC)：The enhancement of the rate of a spin-forbidden process by the presence of an atom of high atomic number, which is either part of, or external to, the excited molecular entity. Mechanistically, it responds to a spin-orbit coupling enhancement produced by a heavy atom.

(1) 《原子物理》杨家福
(2) The Russell Saunders Coupling Scheme-Libretexts
(3) 《固体发光讲义》许少鸿

### 跃迁和选律

(1) 自旋禁戒，不同自旋态($$\Delta S \neq0$$)能级之间的电子跃迁是禁戒的。

(2) 宇称禁戒，如$$d$$层、$$f$$层内或$$d$$层和$$s$$层之间。
(3) Ln3+的$$4f$$电子宇称相同，在自由离子时其电偶极跃迁是禁戒的，而另外两种跃迁(磁偶极和电四级)是允许的，但实验中常能观测到$$4f$$的电偶极跃迁，这是晶体场的影响。

### J-O理论

J-O强度参数可以由稀土的吸收光谱或透射光谱(材料必须透明)+软件计算得到。(待补充)

## 能量传递

### 基质晶格激发

[Debije, Michael. "Better luminescent solar panels in prospect." Nature 519.7543 (2015): 298-299.]

• Förster resonance energy transfer (FRET)
• Coulomb interaction
• Spectral overlap integral  (govern the rate)
• Kappa factor  (the orientation of the two transition dipole moments involved in the process)
• Singlet energy transfer (SEnT) (in general)
• the distance between the donor and the acceptor (typically in the range of 1~10 nm)
• 也有人称作fluorescence resonance energy transfer，但是容易误解，因为能量传递不是transferred by fluorescence，所以Förster resonance energy transfer名字更准确。
• Dexter energy transfer (DET)
• Exchange interaction
• double electron transfer
• electronic coupling (orbital overlap of the of the donor and acceptor)
• triplet energy transfer + short distance SEnT
• short distance的时候可能faster than FRET

Triplet triplet Förster energy transfer, Can this be possible???

#### Förster Energy Transfer Efficiency

$$\phi_{\mathrm{ET}}=\frac{R_{0}^{6}}{R_{0}{ }^{6}+r_{\mathrm{DA}}{ }^{6}}$$$$R_{0} \equiv$$ the Förster distance where $$\phi_{\mathrm{ET}}=0.5$$
$$r_{\mathrm{DA}} \equiv$$ the distance between a donor (fluorescent unit) and an acceptor (可以通过modelling或者some other data来估算)

Förster distance $$\mathrm{R}_{0}(\overset\circ{\mathrm{A}})=0.211 \times\left(\kappa^{2} n^{-4} \phi_{\mathrm{D}} J\right)^{1 / 6}$$$$\kappa^{2}$$: the orientation factor; (orientation of the two chromophores)
$$n$$: the refractive index of the medium;
$$\phi_{\mathrm{D}}$$: the fluorescence quantum yield of the donor;
$$J$$: the overlap integral.

Overlap integral
$$F_{\mathrm{D}}(\lambda)$$: the fluorescence intensity of the donor as a function of wavelength.
$$\varepsilon_{\mathrm{A}}(\lambda)$$: the molar extinction coefficient of the acceptor at that wavelength;

Orientation factor
(1) 不知道的情况下用$$\kappa^2 =2/3$$ (general accepted value，认为donor和acceptor是isotropically oriented)
(2) $$r_{\mathrm{DA}}$$的线connects the centers of the chromophores;
(3) $$\kappa^{2}=\left(\cos \theta_{\mathrm{T}}-3 \cos \theta_{\mathrm{D}} \cos \theta_{\mathrm{A}}\right)^{2}$$
(4) 下图的两个公式是等价的。
(5)$$\kappa=\hat{\mu}_{\mathrm{A}}\cdot \hat{\mu}_{\mathrm{D}}-3\left(\hat{\mu}_{\mathrm{D}}\cdot \hat{R}\right)\left(\hat{\mu}_{\mathrm{A}} \cdot \hat{R}\right)$$其中$$\hat{\mu}_{i}$$ denotes the normalized transition dipole moment of the respective fluorophore, and $$\hat{R}$$ denotes the normalized inter-fluorophore displacement.

(6) Time-dependent analyses of FRET $$k_{\mathrm{ET}}=\left(\frac{R_{0}}{r}\right)^{6} \frac{1}{\tau_{\mathrm{D}}}$$其中$$\tau_{D}$$ is the donor's fluorescence lifetime in the absence of the acceptor，$$R_{0}$$ is the critical distance of Förster radii ($$\phi_{\mathrm{ET}}=0.5$$)，$$r$$是D和A之间的距离。
The FRET efficiency relates to the quantum yield and the fluorescence lifetime of the donor molecule as follows: $$\phi_{\mathrm{ET}}=1-\tau_{\mathrm{D}}^{\prime} / \tau_{\mathrm{D}}$$ where $$\tau_{\mathrm{D}}^{\prime}$$ and $$\tau_{\mathrm{D}}$$ are the donor fluorescence lifetimes in the presence and absence of an acceptor respectively, or as $$\phi_{\mathrm{ET}}=1-F_{\mathrm{D}}^{\prime} / F_{\mathrm{D}}$$ where $$F_{\mathrm{D}}^{\prime}$$ and $$F_{\mathrm{D}}$$ are the donor fluorescence intensities with and without an acceptor respectively.

#### 应用

(1) 基于FRET，利用technique to analyze molecular interactions (e.g. protein-protein interactions). Different fluorophores are used and tagged to proteins. (参考视频)

CFP: Cyan Fluorescent Protein (青色荧光蛋白)
GFP: Green Fluorescent Protein (绿色荧光蛋白)
YFP: Yellow Fluorescent Protein (黄色荧光蛋白)
RFP: Red Fluorescent Protein (红色荧光蛋白) The original was isolated from Discosoma, and named DsRed.

(2) Single-molecule FRET

### 实例—Energy transfer and cross relaxation

Energy transfer processes in Sr3Tb0.90Eu0.10(PO4)3

### 总结

Energy transfer is generally associated with multipolar interactions, radiation reabsorption, or exchange interaction. Among them, Multipolar interactions are usually prevalent which have several types, such as dipole–dipole (d–d), dipole–quadrupole (d–q), and quadrupole–quadrupole (q–q) interactions. Exchange interaction is generally limited to interactions between RE ions in nearest or next nearest neighbor. If migration is rapid compared to direct transfer, quenching tends to be proportional to quenching-ions concentration.

For a better understanding of energy transfer in the host, the relationship of emission intensity and activator concentration is discussed. As the report of Van Uitert and Ozawa, The type of energy transfer can be determined from the change in the emission intensity from the emitting level . The emission intensity  per activator ion follows the equation: $$\frac{I}{x}=K\left[1+\beta(x)^{\theta / 3}\right]^{-1}$$where $$x$$ is the activator concentration, $$I/x$$ is the emission intensity $$(I)$$ per activator concentration $$(x)$$, and $$K$$ and $$\beta$$ are constants for the same excitation condition for a given host crystal.

(1) S首先被激发进入激发态S*
(2) 随后把能量传递给A，使A进入激发态A*，同时S*回到基态S
(3) 处于激发态的中心A*回到基态有两种可能的方式：
(3-a) 通过辐射跃迁的方式回到基态，这种情况我们把S叫作A的敏化剂
(3-b) 通过无辐射跃迁的方式回到基态，这种情况我们把A 叫作S的猝灭剂

### 不同种类中心之间的能量传递

Dexter给出了无辐射ET的几率$$P_{S A}=\left.\frac{2 \pi}{\hbar}\left|<S, A^{*}\right| H_{S A}|S^{*}, A>\right|^{2} \cdot \int g_{s}(E) \cdot g_{A}(E) d E$$其中积分是光谱交叠积分，$$g_S$$是S的发射谱，$$g_A$$是A的吸收谱。矩阵元表示了初态$$\mid S^{*}, A>$$与终态$$\mid S, A^{*}>$$的相互作用贡献，$$H_{SA}$$代表了这种相互作用的哈密顿项。

(1) $$\left|<S, A^{*}\right| H_{S A}\left|S^{*}, A>\right|^{2} \neq 0$$
(2) $$\int g_{s}(E) \cdot g_{A}(E) d E \neq 0$$

(1) 共振，即S的发射谱与A的吸收谱有交叠，交叠积分越大则能量传递几率越大；
(2) 相互作用类型
(2-a) 电多极相互作用，满足$$R^{-n}\,(n=6,8...)$$关系，分别对应电偶极-电偶极跃迁，电偶极-电四级相互作用......
(2-b) 交换相互作用，依赖于波函数的交叠，随距离成指数衰减关系。

• 当实际距离$$R$$大于$$R_c$$时，以S的发射为主；
• 当实际距离$$R$$小于$$R_c$$时，以S向A的能量传递为主。
• 通常对于允许的电偶极相互作用，$$R_c$$约为$$30$$Å；
• 如果偶极跃迁是禁戒的，则需要交换相互作用，其$$R_c$$约为$$5-8$$Å。

S*先发光，然后发出的光被A吸收，导致A的激发和发射。这种传递过程与中心间的距离$$R$$无关。

(1) A的激发谱包含S的特征激发；
(2) 选择对S有效，但是对A无效的激发波长，最后在S和A共掺杂的样品中还是观察到A的发光。

• 带状发射到线谱吸收的$$R_c$$较小，表明ET只能发生在最邻近晶格；
• 线谱发射到带状吸收的$$R_c$$较大，表明ET可以涉及更远的晶格。

(1) Gd3+的$${ }^{6} \mathrm{P}_{7 / 2}$$可以传递给大多数稀土离子，除了Pr3+和Tm3+，因为没有光谱交叠(Dieke diagram)，不满足共振条件。
(2) Ca5(PO4)3F:Sb3+,Mn2+的发光， Sb3+传递能量 给Mn2+， Sb3+的发射谱覆盖了几个Mn2+的吸收 带。由于f值很低(自旋宇称禁戒)，能量传递 的相互作用属于交换相互作用$$R_{c} \sim 7$$Å。
(3) Rb2ZnBr4:Eu2+， Eu2+占据Rb+位置，因为Rb+有两个不同格位，因此Eu2+也有两种不同格 位， Eu2+受到晶格场的影响强烈，两个Eu2+具 有不同的能量。 能量传递可以从高能级的Eu2+(415nm)到低 能级的Eu2+(435nm)。 都属于电偶极允许的跃迁， $$R_{c}=$$35 Å。

### 同种中心之间的能量传递

(1) 当浓度较大时： 中心间的距离小于临界距离，它们就 会产生级联能量传递，即从一个中心传递到下一个中 心，再到下一个中心。。。。。。（发生能量迁移） 直到最后进入一个猝灭中心，导致发光的猝灭，我们 把这种猝灭叫做浓度猝灭。
(2) 当浓度较小时： 这种级联能量传递过程受到阻碍，可以产生发光。
(3) 实验观察： 实验上如果做出一条发光强度随掺杂浓度变化的曲线，我们会观察到开始随着浓度的增加发光强度逐渐增强，到达某个浓度值以后发光强度开始逐渐下降，这就是浓度猝灭开始产生作用。

### 半导体中的能量传递

• 半导体的激发涉及到产生自由电子或空穴。或电子-空穴的束缚状态——激子。
• 自由电子和空穴沿着同一方向运动，直到被辐射跃迁中心俘获，产生发光。
• 激子：包括Frenkel和Wannier激子，属于电子空穴的束缚态。它们可以一起移动，产生能量的输运。自由激子可能在某些位置被束缚，形成束缚激子，束缚激子可能产生发光，也可能发生无辐射跃迁。

## Inokuti-Hirayama

Photoluminescence properties of Eu2+-activated Sr3SiO5 phosphors

https://www.koushare.com/video/videodetail/9291

## 稀土掺杂光量子材料

(1) Defect Engineering for Quantum Grade Rare-Earth Nanocrystals-ACS Nano-2020
(2) Philippe Goldner 视频